טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentBahat-Treidel Omri
SubjectK-Space Singularities in Nonlinear Wave Systems
DepartmentDepartment of Physics
Supervisor ? 18? Mordechai Segev
Full Thesis textFull thesis text - English Version


Abstract

In the following dissertation I describe my study of linear and nonlinear wave dynamics in honeycomb lattices. As opposed to most periodic systems, the band structure of the honeycomb lattice is singular in two different points. In the vicinity of these singularities, the energy manifold, E(k), is a cone known in the literature as the Dirac cone, since the Bloch waves that reside in the vicinity of these singularities are very well described by the massless Dirac Hamiltonian. However, the dispersion relation alone does not capture the entire uniqueness of the system: since the honeycomb lattice is not a Bravais lattice but rather it is made of two trigonal sub lattices. Therefore, the Bloch waves of the system have two components that describe the amplitude of the wave in each sub-lattice. This additional degree of freedom is often referred to as pseudo-spin. The Dirac equation describing the waves is written in the pseudo-spin space, not the real spin, and is therefore relevant for describing bosons such as electro-magnetic waves in periodic media and cold atoms in an optical lattice. In my research, I put a great deal of emphasis on deformed honeycomb lattice, i.e., breaking the C3v symmetry of the lattice. The symmetry can be broken simply by applying strain to a graphene sheet, modifying the amplitude of one of the beams that generate the optical lattice for cold atoms, or changing the distance between waveguides. When the C3v symmetry is broken, the Dirac points move toward each other, eventually annihilate each other and form a gap. Under such circumstances, the effective Dirac Hamiltonian is no longer valid and it is extremely anisotropic. In addition, when the two cones are not distinct another degree of freedom (valley) is lost, and as a consequence some of the ”missing” Hall plateaus should appear. I have focused on studying the nonlinear evolution of wave-packets comprising of waves that reside close to the Dirac points in deformed lattices. To be more specific, I have been studying the modification of conical diffraction when nonlinearity and deformations are applied to the lattice. In addition, I studied Klein tunneling in deformed lattices in the Dirac approximation and beyond it in the both linear and the nonlinear regime.